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## G = C13×C22⋊Q8order 416 = 25·13

### Direct product of C13 and C22⋊Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C13×C22⋊Q8
 Chief series C1 — C2 — C22 — C2×C26 — C2×C52 — Q8×C26 — C13×C22⋊Q8
 Lower central C1 — C22 — C13×C22⋊Q8
 Upper central C1 — C2×C26 — C13×C22⋊Q8

Generators and relations for C13×C22⋊Q8
G = < a,b,c,d,e | a13=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 100 in 74 conjugacy classes, 48 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], Q8 [×2], C23, C13, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, C26 [×3], C26 [×2], C22⋊Q8, C52 [×2], C52 [×5], C2×C26, C2×C26 [×2], C2×C26 [×2], C2×C52 [×2], C2×C52 [×4], C2×C52 [×2], Q8×C13 [×2], C22×C26, C13×C22⋊C4 [×2], C13×C4⋊C4, C13×C4⋊C4 [×2], C22×C52, Q8×C26, C13×C22⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, C13, C2×D4, C2×Q8, C4○D4, C26 [×7], C22⋊Q8, C2×C26 [×7], D4×C13 [×2], Q8×C13 [×2], C22×C26, D4×C26, Q8×C26, C13×C4○D4, C13×C22⋊Q8

Smallest permutation representation of C13×C22⋊Q8
On 208 points
Generators in S208
(1 2 3 4 5 6 7 8 9 10 11 12 13)(14 15 16 17 18 19 20 21 22 23 24 25 26)(27 28 29 30 31 32 33 34 35 36 37 38 39)(40 41 42 43 44 45 46 47 48 49 50 51 52)(53 54 55 56 57 58 59 60 61 62 63 64 65)(66 67 68 69 70 71 72 73 74 75 76 77 78)(79 80 81 82 83 84 85 86 87 88 89 90 91)(92 93 94 95 96 97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112 113 114 115 116 117)(118 119 120 121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140 141 142 143)(144 145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168 169)(170 171 172 173 174 175 176 177 178 179 180 181 182)(183 184 185 186 187 188 189 190 191 192 193 194 195)(196 197 198 199 200 201 202 203 204 205 206 207 208)
(1 122)(2 123)(3 124)(4 125)(5 126)(6 127)(7 128)(8 129)(9 130)(10 118)(11 119)(12 120)(13 121)(14 174)(15 175)(16 176)(17 177)(18 178)(19 179)(20 180)(21 181)(22 182)(23 170)(24 171)(25 172)(26 173)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 79)(50 80)(51 81)(52 82)(53 154)(54 155)(55 156)(56 144)(57 145)(58 146)(59 147)(60 148)(61 149)(62 150)(63 151)(64 152)(65 153)(92 196)(93 197)(94 198)(95 199)(96 200)(97 201)(98 202)(99 203)(100 204)(101 205)(102 206)(103 207)(104 208)(105 141)(106 142)(107 143)(108 131)(109 132)(110 133)(111 134)(112 135)(113 136)(114 137)(115 138)(116 139)(117 140)(157 192)(158 193)(159 194)(160 195)(161 183)(162 184)(163 185)(164 186)(165 187)(166 188)(167 189)(168 190)(169 191)
(1 84)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 91)(9 79)(10 80)(11 81)(12 82)(13 83)(14 116)(15 117)(16 105)(17 106)(18 107)(19 108)(20 109)(21 110)(22 111)(23 112)(24 113)(25 114)(26 115)(27 190)(28 191)(29 192)(30 193)(31 194)(32 195)(33 183)(34 184)(35 185)(36 186)(37 187)(38 188)(39 189)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 129)(49 130)(50 118)(51 119)(52 120)(53 98)(54 99)(55 100)(56 101)(57 102)(58 103)(59 104)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 162)(67 163)(68 164)(69 165)(70 166)(71 167)(72 168)(73 169)(74 157)(75 158)(76 159)(77 160)(78 161)(131 179)(132 180)(133 181)(134 182)(135 170)(136 171)(137 172)(138 173)(139 174)(140 175)(141 176)(142 177)(143 178)(144 205)(145 206)(146 207)(147 208)(148 196)(149 197)(150 198)(151 199)(152 200)(153 201)(154 202)(155 203)(156 204)
(1 160 122 195)(2 161 123 183)(3 162 124 184)(4 163 125 185)(5 164 126 186)(6 165 127 187)(7 166 128 188)(8 167 129 189)(9 168 130 190)(10 169 118 191)(11 157 119 192)(12 158 120 193)(13 159 121 194)(14 63 139 199)(15 64 140 200)(16 65 141 201)(17 53 142 202)(18 54 143 203)(19 55 131 204)(20 56 132 205)(21 57 133 206)(22 58 134 207)(23 59 135 208)(24 60 136 196)(25 61 137 197)(26 62 138 198)(27 79 72 49)(28 80 73 50)(29 81 74 51)(30 82 75 52)(31 83 76 40)(32 84 77 41)(33 85 78 42)(34 86 66 43)(35 87 67 44)(36 88 68 45)(37 89 69 46)(38 90 70 47)(39 91 71 48)(92 171 148 113)(93 172 149 114)(94 173 150 115)(95 174 151 116)(96 175 152 117)(97 176 153 105)(98 177 154 106)(99 178 155 107)(100 179 156 108)(101 180 144 109)(102 181 145 110)(103 182 146 111)(104 170 147 112)
(1 26 122 138)(2 14 123 139)(3 15 124 140)(4 16 125 141)(5 17 126 142)(6 18 127 143)(7 19 128 131)(8 20 129 132)(9 21 130 133)(10 22 118 134)(11 23 119 135)(12 24 120 136)(13 25 121 137)(27 102 72 145)(28 103 73 146)(29 104 74 147)(30 92 75 148)(31 93 76 149)(32 94 77 150)(33 95 78 151)(34 96 66 152)(35 97 67 153)(36 98 68 154)(37 99 69 155)(38 100 70 156)(39 101 71 144)(40 172 83 114)(41 173 84 115)(42 174 85 116)(43 175 86 117)(44 176 87 105)(45 177 88 106)(46 178 89 107)(47 179 90 108)(48 180 91 109)(49 181 79 110)(50 182 80 111)(51 170 81 112)(52 171 82 113)(53 164 202 186)(54 165 203 187)(55 166 204 188)(56 167 205 189)(57 168 206 190)(58 169 207 191)(59 157 208 192)(60 158 196 193)(61 159 197 194)(62 160 198 195)(63 161 199 183)(64 162 200 184)(65 163 201 185)

G:=sub<Sym(208)| (1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140,141,142,143)(144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169)(170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195)(196,197,198,199,200,201,202,203,204,205,206,207,208), (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,118)(11,119)(12,120)(13,121)(14,174)(15,175)(16,176)(17,177)(18,178)(19,179)(20,180)(21,181)(22,182)(23,170)(24,171)(25,172)(26,173)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,79)(50,80)(51,81)(52,82)(53,154)(54,155)(55,156)(56,144)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(92,196)(93,197)(94,198)(95,199)(96,200)(97,201)(98,202)(99,203)(100,204)(101,205)(102,206)(103,207)(104,208)(105,141)(106,142)(107,143)(108,131)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(157,192)(158,193)(159,194)(160,195)(161,183)(162,184)(163,185)(164,186)(165,187)(166,188)(167,189)(168,190)(169,191), (1,84)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,79)(10,80)(11,81)(12,82)(13,83)(14,116)(15,117)(16,105)(17,106)(18,107)(19,108)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,190)(28,191)(29,192)(30,193)(31,194)(32,195)(33,183)(34,184)(35,185)(36,186)(37,187)(38,188)(39,189)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,118)(51,119)(52,120)(53,98)(54,99)(55,100)(56,101)(57,102)(58,103)(59,104)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,162)(67,163)(68,164)(69,165)(70,166)(71,167)(72,168)(73,169)(74,157)(75,158)(76,159)(77,160)(78,161)(131,179)(132,180)(133,181)(134,182)(135,170)(136,171)(137,172)(138,173)(139,174)(140,175)(141,176)(142,177)(143,178)(144,205)(145,206)(146,207)(147,208)(148,196)(149,197)(150,198)(151,199)(152,200)(153,201)(154,202)(155,203)(156,204), (1,160,122,195)(2,161,123,183)(3,162,124,184)(4,163,125,185)(5,164,126,186)(6,165,127,187)(7,166,128,188)(8,167,129,189)(9,168,130,190)(10,169,118,191)(11,157,119,192)(12,158,120,193)(13,159,121,194)(14,63,139,199)(15,64,140,200)(16,65,141,201)(17,53,142,202)(18,54,143,203)(19,55,131,204)(20,56,132,205)(21,57,133,206)(22,58,134,207)(23,59,135,208)(24,60,136,196)(25,61,137,197)(26,62,138,198)(27,79,72,49)(28,80,73,50)(29,81,74,51)(30,82,75,52)(31,83,76,40)(32,84,77,41)(33,85,78,42)(34,86,66,43)(35,87,67,44)(36,88,68,45)(37,89,69,46)(38,90,70,47)(39,91,71,48)(92,171,148,113)(93,172,149,114)(94,173,150,115)(95,174,151,116)(96,175,152,117)(97,176,153,105)(98,177,154,106)(99,178,155,107)(100,179,156,108)(101,180,144,109)(102,181,145,110)(103,182,146,111)(104,170,147,112), (1,26,122,138)(2,14,123,139)(3,15,124,140)(4,16,125,141)(5,17,126,142)(6,18,127,143)(7,19,128,131)(8,20,129,132)(9,21,130,133)(10,22,118,134)(11,23,119,135)(12,24,120,136)(13,25,121,137)(27,102,72,145)(28,103,73,146)(29,104,74,147)(30,92,75,148)(31,93,76,149)(32,94,77,150)(33,95,78,151)(34,96,66,152)(35,97,67,153)(36,98,68,154)(37,99,69,155)(38,100,70,156)(39,101,71,144)(40,172,83,114)(41,173,84,115)(42,174,85,116)(43,175,86,117)(44,176,87,105)(45,177,88,106)(46,178,89,107)(47,179,90,108)(48,180,91,109)(49,181,79,110)(50,182,80,111)(51,170,81,112)(52,171,82,113)(53,164,202,186)(54,165,203,187)(55,166,204,188)(56,167,205,189)(57,168,206,190)(58,169,207,191)(59,157,208,192)(60,158,196,193)(61,159,197,194)(62,160,198,195)(63,161,199,183)(64,162,200,184)(65,163,201,185)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140,141,142,143)(144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169)(170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195)(196,197,198,199,200,201,202,203,204,205,206,207,208), (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,118)(11,119)(12,120)(13,121)(14,174)(15,175)(16,176)(17,177)(18,178)(19,179)(20,180)(21,181)(22,182)(23,170)(24,171)(25,172)(26,173)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,79)(50,80)(51,81)(52,82)(53,154)(54,155)(55,156)(56,144)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(92,196)(93,197)(94,198)(95,199)(96,200)(97,201)(98,202)(99,203)(100,204)(101,205)(102,206)(103,207)(104,208)(105,141)(106,142)(107,143)(108,131)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(157,192)(158,193)(159,194)(160,195)(161,183)(162,184)(163,185)(164,186)(165,187)(166,188)(167,189)(168,190)(169,191), (1,84)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,79)(10,80)(11,81)(12,82)(13,83)(14,116)(15,117)(16,105)(17,106)(18,107)(19,108)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,190)(28,191)(29,192)(30,193)(31,194)(32,195)(33,183)(34,184)(35,185)(36,186)(37,187)(38,188)(39,189)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,118)(51,119)(52,120)(53,98)(54,99)(55,100)(56,101)(57,102)(58,103)(59,104)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,162)(67,163)(68,164)(69,165)(70,166)(71,167)(72,168)(73,169)(74,157)(75,158)(76,159)(77,160)(78,161)(131,179)(132,180)(133,181)(134,182)(135,170)(136,171)(137,172)(138,173)(139,174)(140,175)(141,176)(142,177)(143,178)(144,205)(145,206)(146,207)(147,208)(148,196)(149,197)(150,198)(151,199)(152,200)(153,201)(154,202)(155,203)(156,204), (1,160,122,195)(2,161,123,183)(3,162,124,184)(4,163,125,185)(5,164,126,186)(6,165,127,187)(7,166,128,188)(8,167,129,189)(9,168,130,190)(10,169,118,191)(11,157,119,192)(12,158,120,193)(13,159,121,194)(14,63,139,199)(15,64,140,200)(16,65,141,201)(17,53,142,202)(18,54,143,203)(19,55,131,204)(20,56,132,205)(21,57,133,206)(22,58,134,207)(23,59,135,208)(24,60,136,196)(25,61,137,197)(26,62,138,198)(27,79,72,49)(28,80,73,50)(29,81,74,51)(30,82,75,52)(31,83,76,40)(32,84,77,41)(33,85,78,42)(34,86,66,43)(35,87,67,44)(36,88,68,45)(37,89,69,46)(38,90,70,47)(39,91,71,48)(92,171,148,113)(93,172,149,114)(94,173,150,115)(95,174,151,116)(96,175,152,117)(97,176,153,105)(98,177,154,106)(99,178,155,107)(100,179,156,108)(101,180,144,109)(102,181,145,110)(103,182,146,111)(104,170,147,112), (1,26,122,138)(2,14,123,139)(3,15,124,140)(4,16,125,141)(5,17,126,142)(6,18,127,143)(7,19,128,131)(8,20,129,132)(9,21,130,133)(10,22,118,134)(11,23,119,135)(12,24,120,136)(13,25,121,137)(27,102,72,145)(28,103,73,146)(29,104,74,147)(30,92,75,148)(31,93,76,149)(32,94,77,150)(33,95,78,151)(34,96,66,152)(35,97,67,153)(36,98,68,154)(37,99,69,155)(38,100,70,156)(39,101,71,144)(40,172,83,114)(41,173,84,115)(42,174,85,116)(43,175,86,117)(44,176,87,105)(45,177,88,106)(46,178,89,107)(47,179,90,108)(48,180,91,109)(49,181,79,110)(50,182,80,111)(51,170,81,112)(52,171,82,113)(53,164,202,186)(54,165,203,187)(55,166,204,188)(56,167,205,189)(57,168,206,190)(58,169,207,191)(59,157,208,192)(60,158,196,193)(61,159,197,194)(62,160,198,195)(63,161,199,183)(64,162,200,184)(65,163,201,185) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13),(14,15,16,17,18,19,20,21,22,23,24,25,26),(27,28,29,30,31,32,33,34,35,36,37,38,39),(40,41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64,65),(66,67,68,69,70,71,72,73,74,75,76,77,78),(79,80,81,82,83,84,85,86,87,88,89,90,91),(92,93,94,95,96,97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112,113,114,115,116,117),(118,119,120,121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140,141,142,143),(144,145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168,169),(170,171,172,173,174,175,176,177,178,179,180,181,182),(183,184,185,186,187,188,189,190,191,192,193,194,195),(196,197,198,199,200,201,202,203,204,205,206,207,208)], [(1,122),(2,123),(3,124),(4,125),(5,126),(6,127),(7,128),(8,129),(9,130),(10,118),(11,119),(12,120),(13,121),(14,174),(15,175),(16,176),(17,177),(18,178),(19,179),(20,180),(21,181),(22,182),(23,170),(24,171),(25,172),(26,173),(27,72),(28,73),(29,74),(30,75),(31,76),(32,77),(33,78),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,83),(41,84),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,79),(50,80),(51,81),(52,82),(53,154),(54,155),(55,156),(56,144),(57,145),(58,146),(59,147),(60,148),(61,149),(62,150),(63,151),(64,152),(65,153),(92,196),(93,197),(94,198),(95,199),(96,200),(97,201),(98,202),(99,203),(100,204),(101,205),(102,206),(103,207),(104,208),(105,141),(106,142),(107,143),(108,131),(109,132),(110,133),(111,134),(112,135),(113,136),(114,137),(115,138),(116,139),(117,140),(157,192),(158,193),(159,194),(160,195),(161,183),(162,184),(163,185),(164,186),(165,187),(166,188),(167,189),(168,190),(169,191)], [(1,84),(2,85),(3,86),(4,87),(5,88),(6,89),(7,90),(8,91),(9,79),(10,80),(11,81),(12,82),(13,83),(14,116),(15,117),(16,105),(17,106),(18,107),(19,108),(20,109),(21,110),(22,111),(23,112),(24,113),(25,114),(26,115),(27,190),(28,191),(29,192),(30,193),(31,194),(32,195),(33,183),(34,184),(35,185),(36,186),(37,187),(38,188),(39,189),(40,121),(41,122),(42,123),(43,124),(44,125),(45,126),(46,127),(47,128),(48,129),(49,130),(50,118),(51,119),(52,120),(53,98),(54,99),(55,100),(56,101),(57,102),(58,103),(59,104),(60,92),(61,93),(62,94),(63,95),(64,96),(65,97),(66,162),(67,163),(68,164),(69,165),(70,166),(71,167),(72,168),(73,169),(74,157),(75,158),(76,159),(77,160),(78,161),(131,179),(132,180),(133,181),(134,182),(135,170),(136,171),(137,172),(138,173),(139,174),(140,175),(141,176),(142,177),(143,178),(144,205),(145,206),(146,207),(147,208),(148,196),(149,197),(150,198),(151,199),(152,200),(153,201),(154,202),(155,203),(156,204)], [(1,160,122,195),(2,161,123,183),(3,162,124,184),(4,163,125,185),(5,164,126,186),(6,165,127,187),(7,166,128,188),(8,167,129,189),(9,168,130,190),(10,169,118,191),(11,157,119,192),(12,158,120,193),(13,159,121,194),(14,63,139,199),(15,64,140,200),(16,65,141,201),(17,53,142,202),(18,54,143,203),(19,55,131,204),(20,56,132,205),(21,57,133,206),(22,58,134,207),(23,59,135,208),(24,60,136,196),(25,61,137,197),(26,62,138,198),(27,79,72,49),(28,80,73,50),(29,81,74,51),(30,82,75,52),(31,83,76,40),(32,84,77,41),(33,85,78,42),(34,86,66,43),(35,87,67,44),(36,88,68,45),(37,89,69,46),(38,90,70,47),(39,91,71,48),(92,171,148,113),(93,172,149,114),(94,173,150,115),(95,174,151,116),(96,175,152,117),(97,176,153,105),(98,177,154,106),(99,178,155,107),(100,179,156,108),(101,180,144,109),(102,181,145,110),(103,182,146,111),(104,170,147,112)], [(1,26,122,138),(2,14,123,139),(3,15,124,140),(4,16,125,141),(5,17,126,142),(6,18,127,143),(7,19,128,131),(8,20,129,132),(9,21,130,133),(10,22,118,134),(11,23,119,135),(12,24,120,136),(13,25,121,137),(27,102,72,145),(28,103,73,146),(29,104,74,147),(30,92,75,148),(31,93,76,149),(32,94,77,150),(33,95,78,151),(34,96,66,152),(35,97,67,153),(36,98,68,154),(37,99,69,155),(38,100,70,156),(39,101,71,144),(40,172,83,114),(41,173,84,115),(42,174,85,116),(43,175,86,117),(44,176,87,105),(45,177,88,106),(46,178,89,107),(47,179,90,108),(48,180,91,109),(49,181,79,110),(50,182,80,111),(51,170,81,112),(52,171,82,113),(53,164,202,186),(54,165,203,187),(55,166,204,188),(56,167,205,189),(57,168,206,190),(58,169,207,191),(59,157,208,192),(60,158,196,193),(61,159,197,194),(62,160,198,195),(63,161,199,183),(64,162,200,184),(65,163,201,185)])

182 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 13A ··· 13L 26A ··· 26AJ 26AK ··· 26BH 52A ··· 52AV 52AW ··· 52CR order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 size 1 1 1 1 2 2 2 2 2 2 4 4 4 4 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

182 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C13 C26 C26 C26 C26 D4 Q8 C4○D4 D4×C13 Q8×C13 C13×C4○D4 kernel C13×C22⋊Q8 C13×C22⋊C4 C13×C4⋊C4 C22×C52 Q8×C26 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C52 C2×C26 C26 C4 C22 C2 # reps 1 2 3 1 1 12 24 36 12 12 2 2 2 24 24 24

Matrix representation of C13×C22⋊Q8 in GL4(𝔽53) generated by

 42 0 0 0 0 42 0 0 0 0 16 0 0 0 0 16
,
 52 4 0 0 0 1 0 0 0 0 1 0 0 0 0 52
,
 52 0 0 0 0 52 0 0 0 0 52 0 0 0 0 52
,
 30 39 0 0 0 23 0 0 0 0 52 0 0 0 0 52
,
 4 18 0 0 2 49 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(53))| [42,0,0,0,0,42,0,0,0,0,16,0,0,0,0,16],[52,0,0,0,4,1,0,0,0,0,1,0,0,0,0,52],[52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[30,0,0,0,39,23,0,0,0,0,52,0,0,0,0,52],[4,2,0,0,18,49,0,0,0,0,0,1,0,0,1,0] >;

C13×C22⋊Q8 in GAP, Magma, Sage, TeX

C_{13}\times C_2^2\rtimes Q_8
% in TeX

G:=Group("C13xC2^2:Q8");
// GroupNames label

G:=SmallGroup(416,183);
// by ID

G=gap.SmallGroup(416,183);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,624,1273,631,3818]);
// Polycyclic

G:=Group<a,b,c,d,e|a^13=b^2=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
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𝔽